House purchase with unauthorized vestibule

That's true, but we are also planning a few minor structural changes, which are nevertheless subject to approval. As a result, it is of course possible that someone from the authority will come on site and the distance issue would then be obvious.

BayBO, Article 6:

(8) In the determination of the distance areas, the following shall be disregarded

2. subordinate projections such as balconies and single-story bay windows, if they
a) in total occupy no more than one third of the width of the outer wall of the respective building, at most 5 m each,
b) do not protrude more than 1.50 m beyond this outer wall, and
c) remain at least 2 m away from the opposite neighboring boundary,

According to your description, a) and b) are met.
c) probably not, as it is only 1.5 m.

Perhaps a subsequent approval can be "arranged" if the neighbor waives a subordinate projection at this point (however this must be explained) and thus the theoretically only 4 m distance between the buildings can be maintained. (There are 3 m on the neighbor’s side and 1.5 m on your side). Possibly also under conditions, such as the construction of a fire wall on the projection.

Staircases, bay windows, and entrances within setback areas as described in the [Bay. Landesbauordnung] are uncritical, they are often seen here. You should only refrain from windows facing the neighbor; if necessary, they might still demand removal at some point if already present. Fire protection is a separate issue.